Newform orbit 847.2.a.f

• Label: 847.2.a.f
• Level: $847$
• Weight: $2$
• Character orbit: 847.a
• Self dual: yes
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{2} + ( - \beta + 1) q^{3} + 3 q^{4} - 2 q^{5} + ( - \beta + 5) q^{6} - q^{7} - \beta q^{8} + ( - 2 \beta + 3) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{4} - 4 q^{5} + 10 q^{6} - 2 q^{7} + 6 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

1.61803
−0.618034

−2.23607

−1.23607

3.00000

−2.00000

2.76393

−1.00000

−2.23607

−1.47214

4.47214

1.2

2.23607

3.23607

3.00000

−2.00000

7.23607

−1.00000

2.23607

7.47214

−4.47214

Atkin-Lehner signs

\( p \)	Sign
\(7\)	\( +1 \)
\(11\)	\( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	847.2.a.f		2
3.b	odd	2	1		7623.2.a.bl		2
7.b	odd	2	1		5929.2.a.m		2
11.b	odd	2	1		77.2.a.d	✓	2
11.c	even	5	2		847.2.f.b		4
11.c	even	5	2		847.2.f.m		4
11.d	odd	10	2		847.2.f.a		4
11.d	odd	10	2		847.2.f.n		4
33.d	even	2	1		693.2.a.h		2
44.c	even	2	1		1232.2.a.m		2
55.d	odd	2	1		1925.2.a.r		2
55.e	even	4	2		1925.2.b.h		4
77.b	even	2	1		539.2.a.f		2
77.h	odd	6	2		539.2.e.i		4
77.i	even	6	2		539.2.e.j		4
88.b	odd	2	1		4928.2.a.bm		2
88.g	even	2	1		4928.2.a.bv		2
231.h	odd	2	1		4851.2.a.y		2
308.g	odd	2	1		8624.2.a.ce		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.a.d	✓	2	11.b	odd	2	1
539.2.a.f		2	77.b	even	2	1
539.2.e.i		4	77.h	odd	6	2
539.2.e.j		4	77.i	even	6	2
693.2.a.h		2	33.d	even	2	1
847.2.a.f		2	1.a	even	1	1	trivial
847.2.f.a		4	11.d	odd	10	2
847.2.f.b		4	11.c	even	5	2
847.2.f.m		4	11.c	even	5	2
847.2.f.n		4	11.d	odd	10	2
1232.2.a.m		2	44.c	even	2	1
1925.2.a.r		2	55.d	odd	2	1
1925.2.b.h		4	55.e	even	4	2
4851.2.a.y		2	231.h	odd	2	1
4928.2.a.bm		2	88.b	odd	2	1
4928.2.a.bv		2	88.g	even	2	1
5929.2.a.m		2	7.b	odd	2	1
7623.2.a.bl		2	3.b	odd	2	1
8624.2.a.ce		2	308.g	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):

\( T_{2}^{2} - 5 \)

\( T_{3}^{2} - 2T_{3} - 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 5 \)
$3$	\( T^{2} - 2T - 4 \)
$5$	\( (T + 2)^{2} \)
$7$	\( (T + 1)^{2} \)
$11$	\( T^{2} \)